The 24-year-old Kleybanova was ranked as high as 20th, but after missing much of the last two seasons because of her illness is now 363rd. She entered the main draw at Flushing Meadows thanks to the WTA's special ranking rule.
Until Monday, she hadn't played in a Grand Slam in 2 1/2 years.
"I'm just taking it step by step, enjoying it every day, trying to improve as much as I can, and play my best tennis," Kleybanova said.
After being diagnosed in the spring of 2011 with Hodgkin's lymphoma, a form of blood cancer, Kleybanova announced she was sick via the WTA's website on July 15 that year—her 22nd birthday. She underwent chemotherapy until December 2011.
"Everything is a little bit new for me right now, so I'm just trying to feel everything and try to deal with things as well as I can for the moment," she said.
The 44th-ranked Puig, who cracked the top 50 after reaching the fourth round at Wimbledon, said it was hard to tell Kleybanova had been ill.
"She was a top-20 for a reason," Puig said. "It's amazing to see how she has come back from that, and be able to play at such a level. Definitely, I won't be surprised to see her at the top."